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Intern  Joined: 13 May 2013
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In the xy-plane , line l intersects a circle with center  [#permalink]

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12 00:00

Difficulty:   45% (medium)

Question Stats: 66% (01:35) correct 34% (01:26) wrong based on 265 sessions

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In the xy-plane, line l intersects a circle with center at origin. Is the slope of line l equal to zero?

(1) Line l passes through second quadrant.

(2) Line l is perpendicular to the tangent of the circle.

Originally posted by innocous on 19 Aug 2013, 10:16.
Last edited by Bunuel on 17 Jul 2014, 10:58, edited 1 time in total.
Edited the OA.
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Re: In the xy-plane , line l intersects a circle with center  [#permalink]

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1
innocous wrote:
In the xy-plane , line l intersects a circle with center at origin.Is the slope of line l equal to zero?

1) line l passes through second quadrant.

2) line l is perpendicular to the tangent of the circle.

ALL LINES WHICH ARE PERPENDICULAR TO TANGENT WILL PASS THROUGH CENTRE.

IMO C.

SEE DIAG FOR EXPLANATION.
Attachments LINE.png [ 20.17 KiB | Viewed 4809 times ]

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Intern  Joined: 15 Aug 2013
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Re: In the xy-plane , line l intersects a circle with center  [#permalink]

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In the xy-plane , line l intersects a circle with center at origin.Is the slope of line l equal to zero?

1) line l passes through second quadrant.
2) line l is perpendicular to the tangent of the circle.

FS1: Line l passing through second quadrant and intersecting the circle can be either parall to x-axis(in which case its slope is 0) or not parallel to x-axis(slope not eual to 0). Hence insufficient.
FS2: Line which is perpendicular to the tangent must pass thought centre (since angle betwwen radius and point of contact of tangent to a circle is 90 degrees). Hence, the line can have multiple possibilities for its slope (including 0 and infinity when the line is parallel to x or y axis respectively). Not sufficient.

Combining the above two, the only possibility is when the slope is negatice but not zero (x-axis is not in II quadrant). Hence slope iof line l is not zero for any case.
Sufficient. So answer should be C.
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Re: In the xy-plane , line l intersects a circle with center  [#permalink]

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zerosleep wrote:
In the xy-plane , line l intersects a circle with center at origin.Is the slope of line l equal to zero?

1) line l passes through second quadrant.
2) line l is perpendicular to the tangent of the circle.

FS1: Line l passing through second quadrant and intersecting the circle can be either parall to x-axis(in which case its slope is 0) or not parallel to x-axis(slope not eual to 0). Hence insufficient.
FS2: Line which is perpendicular to the tangent must pass thought centre (since angle betwwen radius and point of contact of tangent to a circle is 90 degrees). Hence, the line can have multiple possibilities for its slope (including 0 and infinity when the line is parallel to x or y axis respectively). Not sufficient.

Combining the above two, the only possibility is when the slope is negatice but not zero (x-axis is not in II quadrant). Hence slope iof line l is not zero for any case.
Sufficient. So answer should be C.

The 2nd statement does not imply that the line l has to pass through the tangent point. It could be any parallel line perpendicular to the tangent.

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Joined: 02 Sep 2009
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Re: In the xy-plane , line l intersects a circle with center  [#permalink]

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faamir wrote:
zerosleep wrote:
In the xy-plane , line l intersects a circle with center at origin.Is the slope of line l equal to zero?

1) line l passes through second quadrant.
2) line l is perpendicular to the tangent of the circle.

FS1: Line l passing through second quadrant and intersecting the circle can be either parall to x-axis(in which case its slope is 0) or not parallel to x-axis(slope not eual to 0). Hence insufficient.
FS2: Line which is perpendicular to the tangent must pass thought centre (since angle betwwen radius and point of contact of tangent to a circle is 90 degrees). Hence, the line can have multiple possibilities for its slope (including 0 and infinity when the line is parallel to x or y axis respectively). Not sufficient.

Combining the above two, the only possibility is when the slope is negatice but not zero (x-axis is not in II quadrant). Hence slope iof line l is not zero for any case.
Sufficient. So answer should be C.

The 2nd statement does not imply that the line l has to pass through the tangent point. It could be any parallel line perpendicular to the tangent.

You are right the answer must be E:
Attachment: Untitled.png [ 26.48 KiB | Viewed 4149 times ]

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Re: In the xy-plane , line l intersects a circle with center  [#permalink]

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Bunuel,

Thank you so much for the quick response. You are the best :D
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Re: In the xy-plane , line l intersects a circle with center  [#permalink]

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1
Stmt1: If any line l passes through second quadrant and intersects the circle, then it can be
- either parallel to x-axis => slope is 0
- or not parallel to x-axis => slope not equal to 0.
Hence insufficient.

Stmt2: If a line is perpendicular to the tangent must pass thought centre (since angle between radius and point of contact of tangent to a circle is 90 degrees). Hence, the line can have multiple possibilities for its slope (including 0 and infinity when the line is parallel to x or y axis respectively). SO this is also Not sufficient.

If we combine two stmts, the only possibility left is negative slope but not zero (x-axis is not in II quadrant). Hence slope of line will never be 0.
Sufficient. Ans- C.
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Re: In the xy-plane , line l intersects a circle with center  [#permalink]

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Top Contributor
1
innocous wrote:
In the xy-plane, line l intersects a circle with center at origin. Is the slope of line l equal to zero?

(1) Line l passes through second quadrant.

(2) Line l is perpendicular to the tangent of the circle.

Target question: Is the slope of line l equal to zero?

IMPORTANT: Since a circle can have an infinite number of tangent lines, the answer here is E.
To prove this, let's jump straight to . . .

Statements 1 and 2 combined
Here are two possible scenarios that meet the two given conditions:
Case a: Case b: In once case, the slope of line l equals 0, and in the other case, the slope of line l does not equal 0.
Since we cannot answer the target question with certainty, the combined statements are NOT SUFFICIENT

Cheers,
Brent
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Re: In the xy-plane , line l intersects a circle with center  [#permalink]

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_________________ Re: In the xy-plane , line l intersects a circle with center   [#permalink] 18 Jun 2019, 09:57

# In the xy-plane , line l intersects a circle with center  